The chain rule allows us to differentiate a wide variety of functions, but we are able to find antiderivatives for only a limited range of functions. We can sometimes use substitution to rewrite functions in a form that we can integrate.

Example 1:

The variable of integration must match the variable in the expression.

Don’t forget to substitute the value for u back into the problem!

Example:(Exploration 1 in the book) One of the clues that we look for is

if we can find a function and its derivative in the integrand.

The derivative of is .

Note that this only worked because of the 2x in the original.Many integrals can not be done by substitution.

Example 2:

Solve for dx.

Example 3:

Example: (Not in book)

We solve for because we can find it in the integrand.

Example 7:

Example 8: The technique is a little different for definite integrals.

We can find new limits, and then we don’t have to substitute back.

new limit

new limit

We could have substituted back and used the original limits.

Example 8:

Wrong!The limits don’t match!

Using the original limits:

Leave the limits out until you substitute back.

This is usually more work than finding new limits

Example: (Exploration 2 in the book)

Don’t forget to use the new limits.

Separable Differential Equations

A separable differential equation can be expressed as the product of a function of x and a function of y.

Example:

Multiply both sides by dx and divide

both sides by y2 to separate the

variables. (Assume y2 is never zero.)

Separable Differential Equations

A separable differential equation can be expressed as the product of a function of x and a function of y.

Example:

Combined constants of integration

Example 9:

Separable differential equation

Combined constants of integration

Example 9:

We now have y as an implicit

function of x.

We can find y as an explicit function

of x by taking the tangent of both sides.

Notice that we can not factor out the constant C, because the distributive property does not work with tangent.

Until then, remember that half the AP exam and half the nation’s college professors do not allow calculators.

In another generation or so, we might be able to use the calculator to find all integrals.

You must practice finding integrals by hand until you are good at it!

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